Axelsson made several stick moves towards the net on his chance, switching to the backhand just to the right of the crease. The shot hit Blackhawks goaltender Nikolai Khabibulin in the right pad but fluttered into the net just inside the right post.
Marco Sturm scored late in the second period for the Bruins, who have won eight of nine. Thomas made 31 stops for the win.
Jonathan Toews tallied in regulation for the Blackhawks, whose four-game win streak ended. Khabibulin took the loss despite a spectacular 38-save performance.
"I played pretty well, and the guys played pretty hard," Khabibulin said. "As a goalie, that's all you can ask. It's frustrating but there's not much you can do. Just get ready for the next game and I guess we'll have to work on the shootout."
Despite three power play opportunities and 18 shots, Chicago went scoreless in the first period. The best chance for either club came when Bruins forward Chuck Kobasew flew in on a short-handed breakaway midway through, but Khabibulin turned aside the shot with a pad save.
After several good chances on either side, the Bruins jumped on top 1-0 with 53.8 seconds left in the second period. Patrice Bergeron dished from the right point into the right circle for Zdeno Chara, whose shot deflected off Sturm and hit the net on the far side.
Chicago finally broke through on a power play with 10:50 left in regulation. Patrick Kane held the puck on the right wing and sent a shot to the net, where Toews managed to redirect it through Thomas' pads from the top of the crease.
Khabibulin saved the game in overtime, making four saves while Hawks defenseman Cam Barker was off for a hooking call. Chicago forward Patrick Sharp was corralled on a breakaway seconds later and awarded a penalty shot with 2:16 remaining after Thomas made a sprawling glove save.
Thomas preserved the game for Boston by closing off the five-hole and stopping Sharp's chance with his stick.
"The game was enjoyable until the end, when it got very hectic and they threw a penalty shot at me," Thomas said. "By killing those penalties in the first period, it gave us momentum. We seemed to be getting into some exciting games this year, and this was one of them."